Abstract and Applied Analysis

Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

Emin Guliyev, Ahmet Eroglu, and Yagub Mammadov

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Abstract

We consider the generalized shift operator, associated with the Dunkl operator Λ α ( f ) ( x ) = ( d / d x ) f ( x ) + ( ( 2 α + 1 ) / x ) ( ( f ( x ) - f ( - x ) ) / 2 ) , α > - 1 / 2 . We study the boundedness of the Dunkl-type fractional maximal operator M β in the Dunkl-type Morrey space L p , λ , α ( ) , 0 λ < 2 α + 2 . We obtain necessary and sufficient conditions on the parameters for the boundedness M β , 0 β < 2 α + 2 from the spaces L p , λ , α ( ) to the spaces L q , λ , α ( ) , 1 < p q < , and from the spaces L 1 , λ , α ( ) to the weak spaces W L q , λ , α ( ) , 1 < q < . As an application of this result, we get the boundedness of M β from the Dunkl-type Besov-Morrey spaces B p θ , λ , α s ( ) to the spaces B q θ , λ , α s ( ) , 1 < p q < , 0 λ < 2 α + 2 , 1 / p - 1 / q = β / ( 2 α + 2 - λ ) , 1 θ , and 0 < s < 1 .

Article information

Source
Abstr. Appl. Anal. Volume 2010 (2010), Article ID 976493, 10 pages.

Dates
First available in Project Euclid: 1 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1288620738

Digital Object Identifier
doi:10.1155/2010/976493

Mathematical Reviews number (MathSciNet)
MR2660397

Zentralblatt MATH identifier
1210.42035

Citation

Guliyev, Emin; Eroglu, Ahmet; Mammadov, Yagub. Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces. Abstr. Appl. Anal. 2010 (2010), Article ID 976493, 10 pages. doi:10.1155/2010/976493. https://projecteuclid.org/euclid.aaa/1288620738


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